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Mirrors > Home > HSE Home > Th. List > bdophsi | Structured version Visualization version GIF version |
Description: The sum of two bounded linear operators is a bounded linear operator. (Contributed by NM, 9-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nmoptri.1 | ⊢ 𝑆 ∈ BndLinOp |
nmoptri.2 | ⊢ 𝑇 ∈ BndLinOp |
Ref | Expression |
---|---|
bdophsi | ⊢ (𝑆 +op 𝑇) ∈ BndLinOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmoptri.1 | . . . 4 ⊢ 𝑆 ∈ BndLinOp | |
2 | bdopln 29644 | . . . 4 ⊢ (𝑆 ∈ BndLinOp → 𝑆 ∈ LinOp) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 𝑆 ∈ LinOp |
4 | nmoptri.2 | . . . 4 ⊢ 𝑇 ∈ BndLinOp | |
5 | bdopln 29644 | . . . 4 ⊢ (𝑇 ∈ BndLinOp → 𝑇 ∈ LinOp) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ 𝑇 ∈ LinOp |
7 | 3, 6 | lnophsi 29784 | . 2 ⊢ (𝑆 +op 𝑇) ∈ LinOp |
8 | bdopf 29645 | . . . . . 6 ⊢ (𝑆 ∈ BndLinOp → 𝑆: ℋ⟶ ℋ) | |
9 | 1, 8 | ax-mp 5 | . . . . 5 ⊢ 𝑆: ℋ⟶ ℋ |
10 | bdopf 29645 | . . . . . 6 ⊢ (𝑇 ∈ BndLinOp → 𝑇: ℋ⟶ ℋ) | |
11 | 4, 10 | ax-mp 5 | . . . . 5 ⊢ 𝑇: ℋ⟶ ℋ |
12 | 9, 11 | hoaddcli 29551 | . . . 4 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ |
13 | nmopxr 29649 | . . . 4 ⊢ ((𝑆 +op 𝑇): ℋ⟶ ℋ → (normop‘(𝑆 +op 𝑇)) ∈ ℝ*) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ (normop‘(𝑆 +op 𝑇)) ∈ ℝ* |
15 | nmopre 29653 | . . . . 5 ⊢ (𝑆 ∈ BndLinOp → (normop‘𝑆) ∈ ℝ) | |
16 | 1, 15 | ax-mp 5 | . . . 4 ⊢ (normop‘𝑆) ∈ ℝ |
17 | nmopre 29653 | . . . . 5 ⊢ (𝑇 ∈ BndLinOp → (normop‘𝑇) ∈ ℝ) | |
18 | 4, 17 | ax-mp 5 | . . . 4 ⊢ (normop‘𝑇) ∈ ℝ |
19 | 16, 18 | readdcli 10645 | . . 3 ⊢ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ |
20 | nmopgtmnf 29651 | . . . 4 ⊢ ((𝑆 +op 𝑇): ℋ⟶ ℋ → -∞ < (normop‘(𝑆 +op 𝑇))) | |
21 | 12, 20 | ax-mp 5 | . . 3 ⊢ -∞ < (normop‘(𝑆 +op 𝑇)) |
22 | 1, 4 | nmoptrii 29877 | . . 3 ⊢ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇)) |
23 | xrre 12550 | . . 3 ⊢ ((((normop‘(𝑆 +op 𝑇)) ∈ ℝ* ∧ ((normop‘𝑆) + (normop‘𝑇)) ∈ ℝ) ∧ (-∞ < (normop‘(𝑆 +op 𝑇)) ∧ (normop‘(𝑆 +op 𝑇)) ≤ ((normop‘𝑆) + (normop‘𝑇)))) → (normop‘(𝑆 +op 𝑇)) ∈ ℝ) | |
24 | 14, 19, 21, 22, 23 | mp4an 692 | . 2 ⊢ (normop‘(𝑆 +op 𝑇)) ∈ ℝ |
25 | elbdop2 29654 | . 2 ⊢ ((𝑆 +op 𝑇) ∈ BndLinOp ↔ ((𝑆 +op 𝑇) ∈ LinOp ∧ (normop‘(𝑆 +op 𝑇)) ∈ ℝ)) | |
26 | 7, 24, 25 | mpbir2an 710 | 1 ⊢ (𝑆 +op 𝑇) ∈ BndLinOp |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 class class class wbr 5030 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 + caddc 10529 -∞cmnf 10662 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 ℋchba 28702 +op chos 28721 normopcnop 28728 LinOpclo 28730 BndLinOpcbo 28731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-hilex 28782 ax-hfvadd 28783 ax-hvcom 28784 ax-hvass 28785 ax-hv0cl 28786 ax-hvaddid 28787 ax-hfvmul 28788 ax-hvmulid 28789 ax-hvmulass 28790 ax-hvdistr1 28791 ax-hvdistr2 28792 ax-hvmul0 28793 ax-hfi 28862 ax-his1 28865 ax-his2 28866 ax-his3 28867 ax-his4 28868 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-grpo 28276 df-gid 28277 df-ablo 28328 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-nmcv 28383 df-hnorm 28751 df-hba 28752 df-hvsub 28754 df-hosum 29513 df-nmop 29622 df-lnop 29624 df-bdop 29625 |
This theorem is referenced by: bdophdi 29880 nmoptri2i 29882 unierri 29887 |
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